One Axiom to Rule Them All: A Minimalist Axiomatization of Quantiles
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: Quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles—monotonicity, semicontinuity, comonotonic additivity, elicitability, and locality in particular—follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus an axiomatization of VaR is obtained with only one axiom among law-based risk measures.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it